An integral test for the transience of a Brownian path with limited local time

2008/06/03 by Itai Benjamini, Itaï Benjamini, Nathanaël Berestycki +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60G17 #60J65 #60K37 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60G17 #msc:60J65 #msc:60K37

paper · pdf · doi:10.48550/arxiv.0806.0597

3 figures. Some typos corrected.

openalex publication_date 2008/06/03 · arxiv created 2010/04/21 · arxiv updated 2010/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a one-dimensional Brownian motion conditioned on a self-repelling behaviour. Given a nondecreasing positive function f(t), consider the measures mu_t obtained by conditioning a Brownian path so that L_s< f(s), for all s<t, where L_s is the local time spent at the origin by time s. It is shown that the measures mu_t are tight, and that any weak limit of mu_t as t tends to infinity is transient provided that t^-3/2f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems.

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